[math-fun] what is known about the squareful Fibonacci numbers?
[[ apart from the fact that any fib(n) containing a factor p^i (i>1 of course) seems to have mod(n,p)=0 ]] up to fib(258), at least one of fib(k) .. fib(k+6) is squareful. Analogously, the squareful fibs are spaced no more than 6 apart. Anyone for a counter-example? Their separations are counted as {1, 5 times}, {2, 11 times}, {3, 5}, {4, 7}, {5, 4}, {6, 27} So 6 is a ‘preferred distance’ for lowish n. Wouter
up to fib(258), at least one of fib(k) .. fib(k+6) is squareful.
Every sixth Fibonacci number is divisible by F_6 = 8 since the Fibonacci numbers are a strong divisibility sequence, so this will always hold. Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Jul 15, 2012 at 4:00 PM, Wouter Meeussen <wouter.meeussen@telenet.be> wrote:
[[ apart from the fact that any fib(n) containing a factor p^i (i>1 of course) seems to have mod(n,p)=0 ]]
up to fib(258), at least one of fib(k) .. fib(k+6) is squareful. Analogously, the squareful fibs are spaced no more than 6 apart. Anyone for a counter-example?
Their separations are counted as {1, 5 times}, {2, 11 times}, {3, 5}, {4, 7}, {5, 4}, {6, 27} So 6 is a ‘preferred distance’ for lowish n.
Wouter
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Dear all, None of 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, is squareful. On Sun, 15 Jul 2012, Wouter Meeussen wrote:
[[ apart from the fact that any fib(n) containing a factor p^i (i>1 of course) seems to have mod(n,p)=0 ]]
up to fib(258), at least one of fib(k) .. fib(k+6) is squareful. Analogously, the squareful fibs are spaced no more than 6 apart. Anyone for a counter-example?
Their separations are counted as {1, 5 times}, {2, 11 times}, {3, 5}, {4, 7}, {5, 4}, {6, 27} So 6 is a ‘preferred distance’ for lowish n.
Wouter
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2584 is divisible by 4 (In any case, I thought that if a | fib( n ) => a | fib( kn ) for all k, so because fib( 6 ) has a square factor, all fib( 6k ) are squareful (and fib( p ) is a good place to look for big prime factors)) ----- Message from rkg@cpsc.ucalgary.ca --------- Date: Sun, 15 Jul 2012 15:11:30 -0600 (MDT) From: Richard Guy <rkg@cpsc.ucalgary.ca> Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? To: math-fun <math-fun@mailman.xmission.com>
Dear all, None of 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, is squareful.
On Sun, 15 Jul 2012, Wouter Meeussen wrote:
[[ apart from the fact that any fib(n) containing a factor p^i (i>1 of course) seems to have mod(n,p)=0 ]]
up to fib(258), at least one of fib(k) .. fib(k+6) is squareful. Analogously, the squareful fibs are spaced no more than 6 apart. Anyone for a counter-example?
Their separations are counted as {1, 5 times}, {2, 11 times}, {3, 5}, {4, 7}, {5, 4}, {6, 27} So 6 is a ?preferred distance? for lowish n. Wouter
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----- End message from rkg@cpsc.ucalgary.ca -----
I should have also pointed out that sequence A065106 is the sequence of "Smallest Fibonacci index to produce a factor p^2 (for primes p)", so all fib(n) where n is a multiple of an element of this sequence is squareful. ----- Message from mbgreen@cis.upenn.edu --------- Date: Sun, 15 Jul 2012 17:40:11 -0400 From: mbgreen@cis.upenn.edu Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? To: math-fun <math-fun@mailman.xmission.com>, Richard Guy <rkg@cpsc.ucalgary.ca> Cc: "greenwald@cis.upenn.edu" <greenwald@cis.upenn.edu>
2584 is divisible by 4 (In any case, I thought that if a | fib( n ) => a | fib( kn ) for all k, so because fib( 6 ) has a square factor, all fib( 6k ) are squareful (and fib( p ) is a good place to look for big prime factors)) ----- Message from rkg@cpsc.ucalgary.ca --------- Date: Sun, 15 Jul 2012 15:11:30 -0600 (MDT) From: Richard Guy <rkg@cpsc.ucalgary.ca> Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? To: math-fun <math-fun@mailman.xmission.com>
Dear all, None of 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, is squareful.
On Sun, 15 Jul 2012, Wouter Meeussen wrote:
[[ apart from the fact that any fib(n) containing a factor p^i (i>1 of course) seems to have mod(n,p)=0 ]]
up to fib(258), at least one of fib(k) .. fib(k+6) is squareful. Analogously, the squareful fibs are spaced no more than 6 apart. Anyone for a counter-example?
Their separations are counted as {1, 5 times}, {2, 11 times}, {3, 5}, {4, 7}, {5, 4}, {6, 27} So 6 is a ?preferred distance? for lowish n. Wouter
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----- End message from rkg@cpsc.ucalgary.ca -----
----- End message from mbgreen@cis.upenn.edu -----
Dear all, I fear that there is confusion about the definition of ``squareful''. See section on powerful numbers in UPINT. R. On Sun, 15 Jul 2012, mbgreen@cis.upenn.edu wrote:
I should have also pointed out that sequence A065106 is the sequence of "Smallest Fibonacci index to produce a factor p^2 (for primes p)", so all fib(n) where n is a multiple of an element of this sequence is squareful. ----- Message from mbgreen@cis.upenn.edu --------- Date: Sun, 15 Jul 2012 17:40:11 -0400 From: mbgreen@cis.upenn.edu Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? To: math-fun <math-fun@mailman.xmission.com>, Richard Guy <rkg@cpsc.ucalgary.ca> Cc: "greenwald@cis.upenn.edu" <greenwald@cis.upenn.edu>
2584 is divisible by 4 (In any case, I thought that if a | fib( n ) => a | fib( kn ) for all k, so because fib( 6 ) has a square factor, all fib( 6k ) are squareful (and fib( p ) is a good place to look for big prime factors)) ----- Message from rkg@cpsc.ucalgary.ca --------- Date: Sun, 15 Jul 2012 15:11:30 -0600 (MDT) From: Richard Guy <rkg@cpsc.ucalgary.ca> Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? To: math-fun <math-fun@mailman.xmission.com>
Dear all, None of 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, is squareful.
On Sun, 15 Jul 2012, Wouter Meeussen wrote:
[[ apart from the fact that any fib(n) containing a factor p^i (i>1 of course) seems to have mod(n,p)=0 ]]
up to fib(258), at least one of fib(k) .. fib(k+6) is squareful. Analogously, the squareful fibs are spaced no more than 6 apart. Anyone for a counter-example?
Their separations are counted as {1, 5 times}, {2, 11 times}, {3, 5}, {4, 7}, {5, 4}, {6, 27} So 6 is a ?preferred distance? for lowish n. Wouter
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
----- End message from rkg@cpsc.ucalgary.ca -----
----- End message from mbgreen@cis.upenn.edu -----
Richard Guy probably uses "UPINT" refers to "Unsolved Problems In Number Theory", a book that perhaps defines "powerful number" in a similar way to the following: An integer m such that if p | m, then p^2 | m, is called a powerful number. (which is from http://mathworld.wolfram.com/PowerfulNumber.html ) So the "confusion" would be if some folks thought that only powerful numbers can be "squareful", e.g. 144 could be "squareful" but 2584 would not be because it has 17 as a factor but not 17^2. On 7/15/12, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
Dear all, I fear that there is confusion about the definition of ``squareful''. See section on powerful numbers in UPINT. R.
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
http://oeis.org/wiki/Squareful_numbers "Squareful numbers are numbers for which at least one prime factor exponent is 2, thus are not squarefree numbers, not to be confused with squarefull numbers, numbers for which each prime factor exponent is at least 2. " Elsewhere on the web, the descriptor "square-full" (with - dash) is used, probably in accordance with orthography. Apologies for the confusion, Wouter. -----Original Message----- From: Robert Munafo Sent: Monday, July 16, 2012 5:58 AM To: math-fun Cc: mbgreen@cis.upenn.edu ; greenwald@cis.upenn.edu Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers? Richard Guy probably uses "UPINT" refers to "Unsolved Problems In Number Theory", a book that perhaps defines "powerful number" in a similar way to the following: An integer m such that if p | m, then p^2 | m, is called a powerful number. (which is from http://mathworld.wolfram.com/PowerfulNumber.html ) So the "confusion" would be if some folks thought that only powerful numbers can be "squareful", e.g. 144 could be "squareful" but 2584 would not be because it has 17 as a factor but not 17^2. On 7/15/12, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
Dear all, I fear that there is confusion about the definition of ``squareful''. See section on powerful numbers in UPINT. R.
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Under that definition, 8 would not be squareful. From the example in the referenced webpage, I would guess it's supposed to say "at least one prime factor exponent is at least 2" --ms On 7/16/2012 9:37 AM, Wouter Meeussen wrote:
http://oeis.org/wiki/Squareful_numbers
"Squareful numbers are numbers for which at least one prime factor exponent is 2, thus are not squarefree numbers, not to be confused with squarefull numbers, numbers for which each prime factor exponent is at least 2. "
Elsewhere on the web, the descriptor "square-full" (with - dash) is used, probably in accordance with orthography.
Apologies for the confusion,
Wouter.
-----Original Message----- From: Robert Munafo Sent: Monday, July 16, 2012 5:58 AM To: math-fun Cc: mbgreen@cis.upenn.edu ; greenwald@cis.upenn.edu Subject: Re: [math-fun] what is known about the squareful Fibonacci numbers?
Richard Guy probably uses "UPINT" refers to "Unsolved Problems In Number Theory", a book that perhaps defines "powerful number" in a similar way to the following:
An integer m such that if p | m, then p^2 | m, is called a powerful number.
(which is from http://mathworld.wolfram.com/PowerfulNumber.html )
So the "confusion" would be if some folks thought that only powerful numbers can be "squareful", e.g. 144 could be "squareful" but 2584 would not be because it has 17 as a factor but not 17^2.
On 7/15/12, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
Dear all, I fear that there is confusion about the definition of ``squareful''. See section on powerful numbers in UPINT. R.
participants (6)
-
Charles Greathouse -
mbgreen＠cis.upenn.edu -
Mike Speciner -
Richard Guy -
Robert Munafo -
Wouter Meeussen